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Eigenvalues and eigenvectors of a Hecke operator on $$S^2$$. (Valeurs propres et vecteurs propres d’un opérateur de Hecke sur $$S^2$$.) (French) Zbl 0900.11033
Séminaire de théorie spectrale et géométrie. Année 1986-1987. Chambéry: Univ. de Savoie, Fac. des Sciences, Service de Mathématiques, Sémin. Théor. Spectrale Géom., Chambéry-Grenoble. 5, 165-173 (1987).
The author determines some eigenvalues and eigenvectors associated to the Hecke operator $$T_5$$ used by A. Lubotzky, R. Phillips, and P. Sarnak [Commun. Pure Appl. Math. 40, 401-420 (1987; Zbl 0648.10034) and ibid. 39, Suppl., S149–S186 (1986; Zbl 0619.10052)]. $$T_5$$ acts stably on $$H_n$$, the space of harmonic spheres of degree $$n$$. The author demonstrates that, while for $$n\leq 5$$ the eigenvalues are necessarily rational, this is not the case for $$n\geq 6$$.
For the entire collection see [Zbl 0825.00040].
##### MSC:
 11F72 Spectral theory; trace formulas (e.g., that of Selberg) 11F55 Other groups and their modular and automorphic forms (several variables) 11P21 Lattice points in specified regions
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